$ 1.\overline{7} \div 2.\overline{8} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 17.7777...\\ x &= 1.7777...\end{align*} $ $\begin{align*} 9x &= 16 \\ x &= \dfrac{16}{9}\end{align*} $ $\begin{align*} 10y &= 28.8888...\\ y &= 2.8888...\end{align*} $ $\begin{align*} 9y &= 26 \\ y &= \dfrac{26}{9}\end{align*} $ So, the problem becomes: $ \dfrac{16}{9} \div \dfrac{26}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{16}{9} \times \dfrac{9}{26} = {?} $ $ \phantom{\dfrac{16}{9} \times \dfrac{26}{9}} = \dfrac{16 \times 9}{9 \times 26} $ $ \phantom{\dfrac{16}{9} \times \dfrac{26}{9}} = \dfrac{16 \times \cancel{9}} {\cancel{9} \times 26} $ $ \phantom{\dfrac{16}{9} \times \dfrac{26}{9}} = \dfrac{16}{26} $ Simplify: ${= \dfrac{8}{13}}$